A less conservative phaselock criterion with linear matrix inequality condition

Frequency-based Popov criterion has been previously used to analyse and design the analog phase-locked loops (PLLs) in the literature. Although in general it is better than the circle criterion for Lur'e systems with sector-bounded nonlinearities, Popov criterion may be conservative when the multiplier is limited to be nonnegative. Furthermore, for high order systems or multi-input-multi-output cases, the frequency-based approach will be computationally intensive. In this paper, a less conservative phaselock condition based on extended Popov criterion where the multiplier is indefinite is presented. The result is formulated in terms of linear matrix inequality which can be easily solved via convex optimization methods. A numerical example with Butterworth filter is provided to show that the result provides a significant improvement for the stability and locking frequency of analog PLLs.